So long as your outfit is a suitable match and your nose weight more or less as it should be, you should be fine. If the wind at the bridge is too strong they won't let you cross, but this does not happen very often.
Keep an eye on flags, smoke etc. so you know which way the wind is coming from and if very strong then reduce your speed. HGVs passing can be disturbing if the wind is coming from your right hand side as you get a short period of 'shelter' when they are alongside.
As you say, the whole question is a variable as a piece of string, but with a well matched and well loaded outfit I would be ( and am) happy towing in winds up to 40 or 50 mph and have towed in much higher. If you travel any distance into a stong headwind you will notice a considerable increase in fuel consumption - again, slowing down a bit helps.